Sharp Weak Type Estimates for Maximal Operators associated to Rare Bases

Abstract

Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in ${\mathbb{R}}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}}f(x)=\underset{x\in R\in \mathcal{B}}{\sup }\frac{1}{|R|}{\int }_R|f|.$$ We provide a sufficient condition on $\mathcal{B}$ so that the estimate $$|\{x\in {\mathbb{R}}^n:M_{\mathcal{B}}f(x)>\alpha \}|\le C_n{\int }_{{\mathbb{R}}^n}\frac{|f|}{\alpha }{\left(1+{\log }^{+}\frac{|f|}{\alpha }\right)}^{n-1}$$ is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several classes of rare bases including Córdoba, Soria and Zygmund bases.